Periodic Adsorption Processes (PAPs) have gained increasing commercial acceptance as an efficient separation technique for a wide range of applications. These processes consist of vessels or beds packed with solid adsorbent. The adsorbent is brought in contact with a multi-component feed and the separation of the components is based on the difference in the affinity towards the adsorbent particles. Examples of PAPs include vacuum swing adsorption (VSA) to separate oxygen from air, pressure swing adsorption to separate hydrogen from hydrocarbons and simulated moving bed chromatography to separate two enantiomers in the liquid phase. Examples of PAPs include vacuum swing adsorption to separate oxygen from air, pressure swing adsorption to separate hydrogen from hydrocarbons and simulated moving bed chromatography to separate two enantiomers in the liquid phase. Unlike most processes e.g. distillation which operates at steady state conditions, PAPs operate under periodic transient conditions with each bed repeatedly undergoing a sequence of steps. These processes reach a cyclic steady state (CSS), where the concentration profiles change dynamically and the profiles at the beginning of each cycle match with those at the end of the cycle. This CSS operation results in dense constraint Jacobians, where the time required for the computation of the Jacobian and its factorization dominates the overall optimization process.

This talk presents a trust-region SQP algorithm for the solution of minimization problems with non-linear equality constrained problems. Instead of forming and factoring the dense constraint Jacobian, this algorithm approximates the Jacobian of equality constraints with a specialized quasi-Newton method [1]. Hence it is well suited to solve optimization problems related to PAPs. This algorithm represents a Byrd-Omojokun [2] trust-region approach that takes the inexactness of the Jacobian and its null-space representation into account. The global convergence of the algorithm to first-order critical points is ensured by the theoretical results presented in [3]. The quality of the approximated constraint Jacobian can be adjusted by verifying two conditions that measure the inexactness of the null space representation. The two required conditions on the inexactness can be easily verified during the optimization process.

Furthermore, we will discuss briefly how the derivative information is computed. Here we apply a targeted approach [4] combining automatic differentiation and more sophisticated integration algorithms, for example by CVODES, to evaluate the direct sensitivity equation, the adjoint equation and the second order adjoint equation.

Numerical results for a Simulated Moving Bed system and non-isothermal VSA O2 bulk gas separation processes are presented.

References:

1. Griewank, A; Walther, A. On constrained optimization by adjoint-based quasi-Newton methods. Optim. Methods. Softw. 2002; 17: 869-889.

2. Omojokun, E. Trust Region Algorithms for Optimization with Nonlinear Equality and Inequality Constraints, Ph.D. thesis, Department of Computer Science, University of Colorado, 1989.

3. Walther, A. A First-Order Convergence analyis of Trust-Region methods with inexact Jacobians. SIAM J. of Optim, 2008; 19(1): 307-325.

4. Ozyurt, D.B.; Barton, P.I. Cheap Second Order Directional derivatives of stiff ODE Embedded Fuctionals. SIAM J. Sci. Comput. 2005; 26(5): 1725-1743.